Take the following sentence with punctuation purposefully omitted:
Do we have an Italian restaurant right next to the Video Outpost "Too"
Video Outpost "Too" is a proper title, specifically the name of a store. In American English, we are taught that when ending a quote, if said quote is at the end of a sentence, the final punctuation goes inside of the ending quotation mark, such as in the example below.
He said, "today is a nice day."
However, does this rule still apply when the quotation marks are not part of a quote but are a stylistic choice on the part of the name? More specifically, which of the two forms below is correct or most correct?
Do we have an Italian restaurant right next to the Video Outpost "Too"?
Do we have an Italian restaurant right next to the Video Outpost "Too?"
Forgive me if this is a duplicate. I've found questions regarding actual quotes, but nothing for this particular situation.
Edit: I would appreciate answers for both the American and British conventions, if possible. I believe this question will be of help to a wider audience if both conventions are provided, assuming there are multiple conventions.
Edit 2: As TeacherKSHuang noted, this is similar to the question of whether punctuation goes inside or outside of quotation marks, however the linked question only covers cases when the quotation marks are used as markers of a quotation. This specific question refers to quote marks that are part of a proper name and are not used as quotation marks.